GAIA Seminar on Complex Analytic Geometry- Dr. Dao Phoung Bac

Abstract: Let $G$ be a reductive group acting linearly on the vector space $V$ via representation $\rho: G \mtn \GL(V)$ defined over an algebraically closed field $k$, and let $v \in V$ be a semistable point, i.e., $0 \notin \overline{G.v}$. Hilbert-Mumford Theorem (1965) characterized an useful criterion for semistable points, namely, there exists a cocharacter $\lambda \in X_{*}(G)$ such that $\lim_{\al \to 0} \lambda(\al).v=0$. The studying of semistable points is motivated by the determining of quotient of an algebraic variety under the action of reductive groups. Furthermore, in 1978, G. Kempf and G. Rousseau(independently) improved this remarkable result by showing that there exists a so-called optimal cocharacter $\lambda_{v}$ satisfying $\lim_{\al \to 0}\lambda_{v}(\al).v \in \overline{G.v} \setminus (G.v)$ and $\lambda_{v}$ takes $v$ outside $G.v$ fastest in some sense. This allows us to deal with many problems of geometric invariant theory over perfect (but non-algebraically closed) base fields. In this talk, we present some refinements and applications of these results for rationality problem of orbits.